Abstract

Using interval Taylor models (TM), we construct algorithms for the computer-assisted proof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs). Although TMs allow one to construct guaranteed estimates for families of solutions of systems of ODEs when integrating ODEs over large time intervals, the interval residual included in the TMs begins to grow exponentially and becomes the dominant part of the estimate of the solution pencil, making it practically unusable. To eliminate this deficiency, the creators of the TM—K. Makino and M. Berz—proposed the idea of so-called “shrink wrapping.” We formalize the original algorithm within the framework of the TM definitions we have adopted and propose our own version of the “shrink wrapping,” more accurately adapted to the problem of the computer-aided proof of the existence of periodic trajectories.

中文翻译：

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收缩包装用于区间泰勒模型在常微分方程系统中周期轨迹存在的计算机辅助证明算法中的应用

摘要

使用区间泰勒模型（TM），我们构造了算法，用于计算机辅助证明常微分方程（ODE）系统中周期轨迹的存在。尽管在较长的时间间隔上集成ODE时，TM允许为ODE系统的解系列构造有保证的估计，但是TM中包含的间隔残差开始呈指数增长，并成为解决方案估计的主要部分，从而几乎无法使用。为了消除这种缺陷，TM-K的创建者。Makino和M. Berz提出了所谓的“收缩包装”概念。我们在采用的TM定义框架内将原始算法形式化，并提出了自己的“收缩包装”版本，